examples of semiprimitive rings

The integers Z:
Since ℤ is commutative, any left ideal is two-sided. So the maximal left ideals of ℤ are the maximal ideals of ℤ, which are the ideals p⁢ℤ for p prime.
So J⁢(ℤ)=⋂pp⁢ℤ=(0),
as there are infinitely many primes.